Some Results on Fractional m-Point Boundary Value Problems

In this paper, we will apply some fixed-point theorems to discuss the existence of solutions for fractional m-point boundary value problems D 0 + q u ″ t = h t f u t , t ∈ 0 , 1 , 1 < q ≤ 2 , u ′ 0 = u ″ 0 = u 1 = 0 , u ″ 1 − ∑ i = 1 m − 2 α i u ‴ ξ i = 0 . In addition, we also present Lyapunov’s inequality and Ulam-Hyers stability results for the given m-point boundary value problems.

Объединение классических и динамических неравенств, возникающих при анализе временных масштабов

In this paper, we present an extension of dynamic Renyi’s inequality on time scales by using the time scale Riemann–Liouville type fractional integral. Furthermore, we find generalizations of the well–known Lyapunov’s inequality and Radon’s inequality on time scales by using the time scale Riemann–Liouville type fractional integrals. Our investigations unify and extend some continuous inequalities and their corresponding discrete analogues. В этой статье мы представляем расширение динамического неравенства Реньи на шкалы времени с помощью дробного интеграла типа Римана-Лиувилля. Кроме того, мы находим обобщения хорошо известного неравенства Ляпунова и неравенства Радона на шкалах времени с помощью дробных интегралов типа Римана-Лиувилля на шкале. Наши исследования объединяют и расширяют некоторые непрерывные неравенства и соответствующие им дискретные аналоги.

Analogue of Lyapunov’s inequality for the second-order ordinary differential equation with continuously distributed differentiation operator

For an ordinary second-order differential equation with an operator of continuously distributed differentiation, an analogue of the Lyapunov inequality of the Dirichlet problem is proved.